Question

In a random sample of 400 registered voters, 120 indicated they plan to vote for the Republican Party. Determine a 95% confidence interval for the proportion of all the registered voters who will vote for the Republican Party. ​ Select one: a. (0.255, 0.345) b. (0.275, 0.325) x. (0.295, 0.305) d. Cannot be determined from the information given.

Answer 1

The correct answer is

a. (0.255, 0.345)

Explanation:

In a sample with a number
`n` of people surveyed with a probability of a success of
`\pi`, and a confidence interval
`1-\alpha`, we have the following confidence interval of proportions.

`\pi \pm z\sqrt{(\pi(1-\pi))/(n)}`

In which

Z is the zscore that has a pvalue of
`1 - (\alpha)/(2)`.

For this problem, we have that:

In a random sample of 400 registered voters, 120 indicated they plan to vote for the Republican Party. This means that
`n = 400` and
`\pi = (120)/(400) = 0.3`

Determine a 95% confidence interval for the proportion of all the registered voters who will vote for the Republican Party. ​

So
`\alpha = 0.05`, z is the value of Z that has a pvalue of
`1 - (0.05)/(2) = 0.975`, so
`Z = 1.96`.

The lower limit of this interval is:

`\pi - z\sqrt{(\pi(1-\pi))/(400)} = 0.3 - 1.96\sqrt{(0.3*0.7)/(400)} = 0.255`

The upper limit of this interval is:

`\pi + z\sqrt{(\pi(1-\pi))/(400)} = 0.3 + 1.96\sqrt{(0.3*0.7)/(400)} = 0.345`

The correct answer is

a. (0.255, 0.345)